Rheol Acta &2001) 40: 434±440 Ó Springer-Verlag 2001 ORIGINAL CONTRIBUTION

Burkhardt Dames An empirical model predicting the viscosity Bradley Ron Morrison Norbert Willenbacher of highly concentrated, bimodal dispersions with colloidal interactions

Abstract The relationship between the particles. Starting from a limiting Received: 7 June 2000 Accepted: 12 February 2001 particle size distribution and viscos- value of 2 for non-interacting either ity of concentrated dispersions is of colloidal or non-colloidal particles, great industrial importance, since it e generally increases strongly with is the key to get high solids disper- decreasing particle size. For a given sions or suspensions. The problem is particle system it then can be treated here experimentally as well expressed as a function of the num- as theoretically for the special case ber average particle diameter. As a of strongly interacting colloidal consequence, the viscosity of bimo- particles. An empirical model based dal dispersions varies not only with on a generalized Quemada equation the size ratio of large to small ~ / Àe g ˆ g 1 À / † is used to describe particles, but also depends on the g as a functionmax of volume fraction absolute particle size going through for mono- as well as multimodal a minimum as the size ratio increas- dispersions. The pre-factor g~ ac- es. Furthermore, the well-known counts for the shear rate dependence viscosity minimum for bimodal dis- of g and does not a€ect the shape of persions with volumetric mixing the g vs / curves. It is shown here ratios of around 30/70 of small to for the ®rst time that colloidal large particles is shown to vanish if interactions do not show up in the colloidal interactions contribute maximum packing parameter and signi®cantly. /max can be calculated from the B. Dames á B. R. Morrison particle size distribution without N. Willenbacher &&) further knowledge of the interac- Key words Bimodal dispersions á BASF Aktiengesellschaft tions among the suspended particles. Rheology á Generalized Quemada Polymers Laboratory 67056 Ludwigshafen, Germany On the other hand, the exponent e is equation á Maximum packing e-mail: [email protected] controlled by the interactions among fraction á Colloidal interactions

Introduction reasonable stir-, pump-, and sprayability during trans- port and application. Particle loadings beyond a volume Market competition, cost reduction, and environmental fraction of 0.60 can only be achieved if the particle size driving forces are pushing industrial researchers to distribution is either broad or bi- or multimodal. formulate polymer dispersions with a particle loading Consequently, the rheology and the packing possibilities as high as possible. The bene®t will be an increased time- of highly concentrated suspensions have been discussed space yield during production, reduced transportation intensively by academic as well as industrial researchers. costs, and less energy consumption for the removal of The literature strongly focuses on suspensions of non- solvent in the application process. At the same time the Brownian spheres &e.g., Farris 1968; Chong et al. 1971; viscosity has to be kept low enough in order to ensure a Poslinski et al. 1988; Sengun and Probstein 1989, 1997; sucient heat transfer during polymerization as well as Gondret and Petit 1997) or on Brownian but so-called 435 hard spheres &Rodriguez and Kaler 1992; Chang and size, and that the viscosity minimum for bimodal Powell 1994; Shikata et al. 1998). Only a few papers deal dispersions with volumetric mixing ratios of around with the rheology of Brownian particle suspensions with 30/70 of small to large particles vanishes, if colloidal either electrostatic &Zaman and Moudgil 1999; Richter- interactions contribute signi®cantly. ing and MuÈ ller 1995) or steric interactions &Woutersen and de Kruif 1993; D'Haene and Mewis 1994; Green- wood et al. 1997) or a combination of both &Ho€man Experimental 1992; Chu et al. 1998). Monodisperse suspensions can in principle pack in Samples several ways with di€erent maximum packing fractions, All dispersions were made by seeded emulsion polymerization. For e.g., /m ˆ 0.52 for simple cubic, /m ˆ 0.74 for hexagonal our examinations, we used dispersions with a monomodal as well or 0.63 for random close packing. For a random close as dispersions with a bimodal particle size distribution. In order to packed bimodal system a maximum packing fraction produce bimodal dispersions with high solids content we started the polymerization with two di€erent seeds di€ering in their average /m ˆ 0.87 is achieved assuming Rlarge/Rsmall >10so that small particles together with the solvent molecules particle size. The initial amount of these seeds was governed by the desired particle size distribution. This is a convenient method to can be treated as a continuum. The small particle obtain bimodal dispersions with a very high viscosity because there volume fraction then is ns ˆ 0.27 &Farris 1968). From is no need for mixing two monomodal dispersions and evaporation simple geometrical considerations it follows that the of water afterwards that could induce coagulation and agglomer- small particles exactly ®t into the pores between the large ation. All dispersions had the same chemical composition consist- ing of 99 parts butylacrylate, 1 part acrylic acid, 0.8 parts ones at a critical size ratio Rlarge/Rsmall ˆ 6.46 &McGeary dodecylphenoxybenzene disulfonic acid sodium salt &Dowfax 1961). In this case the number ratio of small to large 2A1), and 0.2 parts sodium laurylsulfate. particles Nsmall/Nlarge  100; that means there are much The initiation was done either with sodiumperoxodisulfate, a so- more small particles than interstices between large called thermal initiation, or with the redox system ascorbic acid/ tert-butylhydroperoxide. Thus we attained dispersions with the particles. Therefore, the simple picture of small particles same particle size distribution but with di€erent surface properties. ®lling the space between the large ones, thus providing a high maximum packing fraction and a low viscosity, is Measurements not valid for bimodal suspensions with size ratios like this. If the size ratio is below this critical value the lattice Viscosity measurements were done with a controlled strain of the large particles may be expanded, and microphase rotational rheometer Rheometrics RFS II equipped with a Couette separation or the formation of superstructures can also &Ri ˆ 16 mm, Ra ˆ 17 mm) sample cell. All measurements were performed at 20 °C. occur. These phenomena have been discussed intensively Particle size distributions were determined using the analytical in the literature &e.g., Verhaegh and Lekkerkerker 1997; ultracentrifuge technique &MaÈ chtle 1992). Typical results are Bartlett and Pusey 1993). Despite the variety of packing presented in Fig. 1, showing the di€erential mass distribution as possibilities, viscosity reduction in bimodal suspensions a function of particle diameter. The mean size of each particle species was determined from the position of the maximum of the is a very general phenomenon and has been observed for corresponding peak, and the relative fraction of each species was large as well as intermediate and small size ratios. A calculated from the area under each peak. minimum viscosity at a small particle volume fraction ns of about 0.3 has been reported for suspensions of non- Brownian glass beads &Chong et al. 1971) as well as for Results and discussion Brownian hard sphere &Rodriguez and Kaler 1992) and colloidally interacting suspensions &Zaman and Moudgil Experiments 1999). The aim of this paper is to work out a procedure The viscosity of a large number of mono- and bimodal which allows for a simple calculation of the viscosity of dispersions was determined as a function of shear rate suspensions or dispersions at particle loadings and shear and volume fraction. Data for a monomodal system rates relevant for industrial applications. An empirical with 250 nm particle diameter are shown in Fig. 2. model based on a generalized Quemada equation is used These results are typical for all suspensions investigated to describe g as a function of volume fraction for mono- here. Varying the particle concentration, two regimes as well as multimodal dispersions. This model quanti- di€ering with respect to the shape of the viscosity curves tatively describes the viscosity of a large number of can be distinguished. bimodal suspensions with signi®cantly di€erent size and At volume fractions / below a critical packing mixing ratios investigated here. Moreover, it qualita- fraction /c the viscosity curves exhibit a Newtonian tively predicts well-known features of suspensions with plateau at low shear rates followed by a shear thinning colloidal interactions. In particular, it is shown, that the region, indicating that the dispersions behave like a viscosity varies not only with the size ratio of large to liquid. The onset of shear thinning is shifted to higher small particles, but also depends on the absolute particle shear rates and the degree of shear thinning decreases 436

Fig. 1 Di€erential particle size distribution from AUC mea- surements for di€erent bimodal dispersions. The labels inserted in the diagrams give Dlarge and Dsmall in nm and the small particle volume fraction times 100 &i.e., B-915/310±10 means Dlarge ˆ 915 nm, Dsmall ˆ 310 nm, and ns  0.10)

Fig. 2 Viscosity vs shear rate for a monomodal dispersion &diameter: 250 nm) at di€erent particle loadings

with decreasing volume fraction. The high shear limiting curves. Therefore, the relative change of viscosity with value of the viscosity is not reached within the shear rate volume fraction is independent of shear rate and it is range &c_ < 500 sÀ1) investigated here. sucient to discuss the e€ect of packing density on For / > /c the dispersions are shear thinning in the viscosity at a ®xed shear rate. whole shear rate range investigated and neither a high Dispersions go through various ¯ow ®elds during nor a low shear plateau is observed. The viscosity curves manufacturing and application; typical shear rates show a slight upward curvature at low shear rates during stirring and pumping are in the range between indicating an apparent yield stress. At these volume 50 s±1 and 500 s±1. From this point of view we have fractions the dispersions show gel-like behavior. The decided to analyze the e€ect of particle size distribution shape of the ¯ow curves hardly changes with the particle on viscosity at a shear rate of c_ ˆ 200 sÀ1. All viscosity density in this regime and the variation of / essentially data shown in the subsequent part of the paper are taken results in a vertical shift of the viscosity vs shear rate at this particular shear rate. 437

 In Fig. 3 the viscosity of a bimodal dispersion with / Àe g ˆ g~ 1 À 3† Rlarge ˆ 270 nm, Rsmall ˆ 88 nm, and a small particle /max volume fraction ns ˆ 0.72 is shown as a function of volume fraction. The pre-factor g~ determines the viscosity level and The particle concentration dependence of the zero depends on shear rate, only. It does not a€ect the shape shear viscosity of liquid-like dispersions is often de- of the g ± / curve. Therefore, it will not be discussed scribed by phenomenological equations like the Krieger- further in the subsequent part of the paper. As Dougherty equation &Krieger and Dougherty 1959): demonstrated in Fig. 3, the generalized Quemada equa- À2:5/ tion provides a very good ®t to the experimental data. / max A large number of mono- and bimodal dispersions g ˆ gsolvent 1 À 1† /max with di€erent absolute particle size, size ratio, and small or the equation derived by Quemada &1977): particle volume fractions have been analyzed in an  analogous way. The resulting values for the maximum / À2 packing fraction /max and the exponent e are discussed g ˆ gsolvent 1 À 2† /max in the next section. As indicated by Eqs &1), &2), and &3) the viscosity of a where g is the viscosity of the dispersing agent or solvent concentrated suspension is strongly controlled by the solvent, which is water in our case, and / is the max ratio of its volume fraction to the maximum packing maximum packing fraction for a given particle size fraction. Consequently, technical improvements aim at distribution. The advantage of the Krieger-Dougherty an increase of / through optimization of the particle equation is, that it reduces to the correct Einstein max size distribution in order to decrease the viscosity at a equation in the limit of in®nite dilution, whereas the given particle loading. McGeary &1961) has performed Quemada equation gives the right divergence for g as / intensive studies regarding the packing properties of approaches / . For a monodisperse suspension with max multimodal suspensions of glass beads and Sudduth / ˆ 0.63 the exponent in Eq. &1) is slightly lower than max &1993) has presented the following mathematical para- in Eq. &2). Trying to ®t both equations to the experi- metrization of these data: mental data in Fig. 3 with / as an adjustable max  parameter shows that neither the Krieger-Dougherty ÀÁ D / ˆ u À u À /mono exp 0:27 1 À 5 4† nor the Quemada equation can match the shape of the max n n max D experimentally determined g ± / curve. Therefore, we 1 introduce an additional phenomenological parameter e with and propose a generalized Quemada equation &Gondrot ÀÁ and Petit 1997): mono n un ˆ 1 À 1 À /max and P n N dx D ˆ iˆ1 i i x Pn xÀ1 iˆ1 Nidi

where /max is the maximum packing fraction of an mono n-modal suspension of spherical particles, /max ˆ 0:63 is the corresponding value for a monodisperse suspen- sion, and Dx is the x-th moment of the particle size distribution, namely D1 is the number average of the particle diameter. We have used Eq. &4) to calculate /max from the particle size distributions of our bimodal suspensions as determined from the analytical ultracentrifuge measure- ments. In Fig. 4 these values are compared to those obtained from ®tting Eq. &3) to the viscosity vs / data. Each data point in that plot represents a particular, separately polymerized suspension. Obviously, there is a strong linear correlation &correlation coecient Fig. 3 Viscosity vs volume fraction for highly concentrated disper- R ˆ 0.95) between these di€erently determined maxi- sions. Generalized Quemada Eq. &3) &Ð Ð Ð), Quemada Eq. &2) mum packing values. This is surprising since the &± ± ±) and Krieger-Dougherty Eq. &1) &¼) ®tted to the experimental á viscosity measurements were performed on dispersions data &d) for a bimodal dispersion with Dlarge ˆ 270 nm, Dsmall ˆ 88 nm, and ns ˆ 0.72 of colloidally interacting Brownian spheres, whereas 438

Eq. &4) describes the packing properties of non-Brown- among the particles. In both cases e starts from a ian hard spheres. Obviously, the packing parameter limiting value of 2 for large mean particle diameters but /max is independent of the interactions among the strongly increases as D1 decreases and values larger than particles and can be predicted from the particle size 4 are obtained at mean particle diameters below 100 nm. distribution alone. This increase is less pronounced for the systems which The values for the exponent e extracted from ®tting were thermally initiated than for those were the Eq. &3) to the g vs / data are shown in Fig. 5 vs the polymerization was started using a redox agent. number average mean particle diameter D1 of the The limiting value of e ˆ 2 reached for D1 > 250 nm corresponding bimodal suspension. Two series of bimo- is in good agreement with the exponent of divergence for dal mixtures have been investigated. These series di€er the viscosity of hard sphere dispersions observed with respect to the initiation of the polymerization which experimentally &De Kruif et al. 1985) as well as predicted is expected to result in a di€erent particle surface theoretically &Brady 1993). This is reasonable since, due structure and hence a di€erent repulsive interaction to the high ionic strength &estimated ion concentra- tion >0.05 mol/l) the range of the electrostatic interac- tions is much shorter than the mean particle separation at the investigated particle volume fractions. Thus, if D1 is large enough these dispersions behave like hard sphere suspensions with respect to the viscosity. On the other hand, the increase of e with decreasing mean particle size is attributed to the e€ect of colloidal interactions. As the mean particle separation decreases, colloidal interac- tions among the particles are getting more and more important and this leads to a much stronger divergence of the viscosity than expected for hard spheres. The di€erences in e observed for the two di€erently poly- merized systems further support the interpretation that the increase of e is governed by colloidal interactions. A dependence of these particle-particle interactions on the structure and chemical composition of the particle surface seems to be reasonable, but a detailed interpre- tation of the relationship between particle interactions Fig. 4 Maximum packing fraction /max as determined from a ®t of Eq. &3) to the g ± / data for various mono- and bimodal dispersions and polymerization conditions is beyond the scope of vs /max as calculated from the particle size distribution according to this work. Eq. &4)

Model calculations

Equation &3) can be used to calculate the viscosity of highly concentrated, multimodal suspensions as a function of volume fraction. The maximum packing fraction /max can be calculated from the particle size distribution according to Eq. &4). The exponent e depends on the colloidal interactions and its dependence on particle size has to be determined from viscosity measurements for a particular type of suspension. We have investigated the e€ect of particle size and mixing ratio on the viscosity of bimodal suspensions. These model calculations are based on the e &D1) data for the thermally initiated dispersions. In order to describe these data numerically, we have ®tted the following Fig. 5 Exponent e as determined from a ®t of Eq. &3) to the g ± / phenomenological equation to the data presented in data for various mono- and bimodal dispersions vs the corresponding Fig. 5: number average of the particle diameter D . Two sets of mixtures ! 1 D 2 polymerized with redox &n)andthermal&d) initiation have been e ˆ e À †e À e exp À 5 5† investigated. The lines correspond to ®ts of Eq. &5) to the experimental 1 0 1 e data D 439

À1 Also the pre-factor g~ c_ ˆ 200 s † which has been decrease in D1) as a consequence of the contribution of determined for this system has been used. colloidal interactions. These results correspond very well The e€ect of particle size ratio on viscosity is shown with experimental data on electrostatically stabilized in Fig. 6. For these calculations Dlarge ˆ 800 nm, a total polystyrene dispersions &Horn 1999). For their system volume fraction / ˆ 0.6, and a small particle volume the well-know viscosity minimum at a volumetric mixing fraction ns ˆ 0.25 have been chosen. Obviously the ratio of around 30/70 of small to large particles is found viscosity strongly decreases as the size of the small at high ionic strength, where the electrostatic particle particles decreases, goes through a minimum at Dlarge/ interactions are strongly suppressed, but vanishes at low Dsmall  4, and then increases again as the size ratio ionic strength, where colloidal interactions contribute further increases. This minimum is a consequence of the signi®cantly. increase of e with decreasing mean particle diameter. If e is kept constant the viscosity decreases monotonically with increasing size ratio, as expected for hard sphere Conclusion suspensions &Chong et al. 1971). In Fig. 6 this is shown for e ˆ 2. The relationship between particle size distribution and The variation of viscosity as a function of small viscosity of highly concentrated, gel-like dispersions has particle volume fraction has been considered for two sets been treated here experimentally as well as theoretically for the special case of strongly interacting colloidal of mixtures with a similar size ratio Dlarge/Dsmall ˆ 3.1 but di€erent absolute particle sizes. The results are particles. An empirical model based on a generalized shown in Fig. 7. In both cases the total volume fraction Quemada equation g ˆ g~ 1 À / †Àe is used to describe /max / was set to 0.6 and the e &D1) data for the thermally g as a function of volume fraction for mono- as well as initiated dispersions were used to perform the calcula- multimodal dispersions. The pre-factor g~ accounts for tions. For the mixture of large particles &680 nm and the shear rate dependence of g and does not a€ect the 210 nm, e ˆ 2 for any value of ns) the viscosity goes shape of the g vs / curves. through a minimum for a small particle volume fraction Colloidal interactions do not show up in the maxi- ns  0.25. This is in accordance with the well-known mum packing parameter. The experimentally deter- behavior of Brownian &Rodriguez and Kaler 1992) as mined /max values agree very well with predictions well as non-Brownian hard sphere suspensions &Chong from an empirical expression derived for non-colloidal et al. 1971; Gondrot and Petit 1997). In contrast, the suspensions with arbitrary particle size distribution. So viscosity increases monotonically with ns for a mixture this parameter can be extracted from the particle size of small particles &250 nm and 80 nm) with a similar size distribution without further knowledge of the nature ratio. This is again due to the fact that e increases with and strength of the interactions among the suspended increasing number of small particles &corresponding to a particles.

Fig. 6 Viscosity vs particle size ratio calculated according to Eq. &3). Fig. 7 Viscosity vs small particle volume fraction ns calculated according to Eq. &3). Model parameters: / ˆ 0.6, / calculated Model parameters: Dlarge ˆ 800 nm, / ˆ 0.6, ns ˆ 0.72, /max calcula- max ted from Eq. &4); &Ð Ð Ð) e &D1)takenfroma®tofEq.&5)tothe from Eq. &4), e&D1) taken from a ®t of Eq. &5) to the experimental data experimental data in Fig. 5 for thermal initiation; &± ± ±) analogous in Fig. 5 for thermal initiation; &Ð Ð Ð) Dlarge ˆ 250 nm, Dsmall ˆ calculation assuming e ˆ 2; &¼¼) dependence of e on the particle size 80 nm, Dlarge/Dsmall ˆ 3.1; &± ± ±) Dlarge ˆ 680 nm, Dsmall ˆ 210 nm, ratio &right y-axis) Dlarge/Dsmall ˆ 3 440

The exponent e is controlled by colloidal interac- only with the size ratio of large to small particles, but tions. Starting from a limiting value of 2 for non- also depends on the absolute particle size going interacting either colloidal or non-colloidal particles, e through a minimum as the size ratio increases. generally increases strongly with decreasing particle Furthermore, the model predicts that the well-known size depending on the range and strength of the viscosity minimum for bimodal dispersions with volu- colloidal interactions. For a given particle system e metric mixing ratios of around 30/70 of small to large then is a function of the number average mean particles vanishes if colloidal interactions contribute diameter D1 only. The presented model describes the signi®cantly. viscosity of a large number of suspensions investigated E€ects like this severely limit the development of here quantitatively. Moreover, it qualitatively predicts colloidal suspensions with high particle loadings. well-known features of bimodal suspensions with colloidal interactions. In particular, it is demonstrated Acknowledgements We thank W MaÈ chtle for performing the that the viscosity of bimodal dispersions varies not particle size analysis and W Richtering for stimulating discussions.

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